\subsection{Convex Hull Intersection}
One problem mentioned in \cite{miller} is that the 6D convex hull created from 6D wrenches is hard to visualize. Futhermore a quality measure given from the size of a 6D wrench looses its physical meaning to some extent. The solution to this problem is to project the convex hull on a 3D space. This can be done by setting either the force or torque to zero. If the torque is set to zero, it is possible to find the force that can be applied when no torque is applied, and vice versa.\\
To illustrate how this is done, the example shown in figure \ref{qhull:initial} is used. 8 vectors span a cube in 3D space, and the convex hull is found by using Qhull\footnote{http://www.qhull.org/}. The intersection with the xy-plane in z=0 is to be found. The principle in the method is the same when doing the intersection between a 6D space and a 3D space, but for simplicity only 3D and 2D is considered.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.5]{figures/initial3D.png}
\caption{Example Problem with a 3D cube that intersects a plane. The sphere has a radius of 0.26.}
\label{qhull:initial}
\end{figure}\\

\noindent It is easy to find the projection of the cube onto the xy-plane, as shown in figure \ref{qhull:projection}. This can be done by simply putting the z coordinate of the 8 vectors to zero, and creating a new convex hull in 2D. The biggest circle fitting inside the convex hull is shown in figure \ref{qhull:projection:2d}. The radius of this circle is 0.69. It is however clear that this projection results in a too optimistic measure of the size of the convex hull. Instead of finding the size of the convex hull when z is zero, the size of the convex hull is found for the most optimistic values of z.
\begin{figure}[htbp]
\centering
\subfigure[Projection in 3D.]{
\includegraphics[scale=0.5]{figures/projection3D.png}
\label{qhull:projection:3d}
}
\subfigure[Projection in 2D.]{
\includegraphics[scale=0.5]{figures/projection2D.png}
\label{qhull:projection:2d}
}
\caption{Projection of the cube.}
\label{qhull:projection}
\end{figure}\\

\noindent By inspiration from the GraspIt! framework\footnote{http://www.cs.columbia.edu/~cmatei/graspit/}, the intersection is found between the convex hull and the plane. A convex hull can be represented by two different representations. Instead of using the vertices, the faces of the convex hull can be used. Each face can be described as a normal vector and the shortest distance to the origo. This representation is illustrated in figure \ref{qhull:intersection:3d}. Each face forms a halfspace, and the convex hull can be seen as a collection of halfspace constraints. Instead of projecting the vertices, the halfspace constraints is projected. This is done by setting the z coordinate of the normal vectors to zero and normalizing the vectors. The offset is changed by dividing the offset by the length of the new normal vector. This gives the distance to the point where the halfspace intersects the xy-plane. The projected halfspaces is shown in figure \ref{qhull:projection:2d} with yellow arrows. The convex hull is then reconstructed by using the "qhalf" program from Qhull, giving the vertices of the interection. The blue triangle is the intersection of the cube with the xy-plane. The radius of the sphere is 0.21.
\begin{figure}[htbp]
\centering
\subfigure[Intersection in 3D.]{
\includegraphics[scale=0.5]{figures/intersection3D.png}
\label{qhull:intersection:3d}
}
\subfigure[Intersection in 2D.]{
\includegraphics[scale=0.5]{figures/intersection2D.png}
\label{qhull:intersection:2d}
}
\caption{Intersection between the convex hull and the plane.}
\label{qhull:intersection}
\end{figure}

\noindent In the 6-dimensional wrench space, one could find the intersection where the torque is zero. This would give the size of the force that can be resisted in all directions, when no torque is applied. No torque is applied only if the gripper grasps the object at exactly the center of mass. In reality this will not hold true. To get a more reliable result, the intersection is found for a worst case wrench. For this, the minimum wrench that can be resisted in 6-dimensional space is found. The torque component of this wrench now gives the worst case torque that the gripper can apply with the specific grasp without loosing the object. The force that can be resisted in all directions when the gripper applies the worst case force is then found from the intersection of the convex hull.